where \(\beta \)  is the inverse temperature and the free energy is \(F(\rho _b)=\mathrm {Tr}(\rho _b\mathcal {F})\)  with \(\mathcal {F}=\mathcal {H}_b+\beta ^{-1}\log \rho _b \) .
Such work can be extracted in single-shot scenarios by correlating the battery to catalytic systems [1]} or in the limit of asymptotically many copies of the battery [2]}.
Another one is the ergotropy which is defined as the most energy that can be extracted by using an optimal cyclic unitary transformation [3]}
\(W_e(t)=\max _{U\in \mathcal {U}_c}\left[ E(\rho _b(t))-E(U\rho _b(t)U^\dag ) \right],\) 
